Group #72244113 — subgroups

Subgroups — C₂×C₉²×D₈ #72244113

Order 2592.

874 conjugacy classes of subgroups; only the 506 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 506 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₉²	C₂	1	2	yes	maximal, complemented
2	C₉²×D₈	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈×C₉²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₉×D₈	C₃	1	4	yes	maximal
5	C₂³×C₉²	—	2	2	no	—
6	D₄×C₉²	—	2	2	no	—
7	D₄×C₉²	C₂²	1	4	yes	complemented
8	C₂×C₄×C₉²	C₂²	1	1	yes	characteristic
9	C₈×C₉²	C₂²	1	2	yes	complemented
10	C₂×C₃×D₄×C₉	C₂×C₃	1	8	yes	—
11	C₃×C₉×D₈	C₂×C₃	1	16	yes	—
12	C₂×C₃×C₈×C₉	C₂×C₃	1	4	yes	—
13	C₂²×C₉²	—	2	4	no	—
14	C₂²×C₉²	—	4	4	no	—
15	C₂²×C₉²	D₄	1	1	yes	characteristic
16	C₄×C₉²	D₄	1	1	yes	characteristic
17	C₄×C₉²	C₂³	1	1	yes	characteristic
18	C₂×C₃²×D₈	C₃²	1	1	yes	characteristic
19	C₂×C₉×D₈	C₉	1	12	yes	direct product factor
20	C₂³×C₃×C₉	—	2	8	no	—
21	C₃×D₄×C₉	—	2	8	no	—
22	C₃×D₄×C₉	C₂²×C₃	1	16	yes	—
23	C₂×C₃×C₄×C₉	C₂²×C₃	1	4	yes	—
24	C₃×C₈×C₉	C₂²×C₃	1	8	yes	—
25	C₂×C₉²	—	4	4	no	—
26	C₂×C₉²	D₈	1	2	yes	direct product factor
27	C₂×C₉²	C₂×D₄	1	1	yes	characteristic
28	C₂×C₃²×D₄	C₂×C₃²	1	2	yes	—
29	C₃²×D₈	C₂×C₃²	1	4	yes	—
30	C₂×C₃²×C₈	C₂×C₃²	1	1	yes	characteristic
31	C₂×D₄×C₉	C₂×C₉	1	24	yes	complemented
32	C₉×D₈	C₂×C₉	1	48	yes	direct product factor
33	C₂×C₈×C₉	C₂×C₉	1	12	yes	complemented
34	C₂²×C₃×C₉	—	2	16	no	—
35	C₂²×C₃×C₉	—	4	16	no	—
36	C₂²×C₃×C₉	C₃×D₄	1	4	yes	—
37	C₃×C₄×C₉	C₃×D₄	1	4	yes	—
38	C₃×C₄×C₉	C₂³×C₃	1	4	yes	—
39	C₂×C₃×D₈	C₃×C₉	1	4	yes	—
40	C₉²	C₂×D₈	1	1	yes	characteristic, direct product factor, Sylow
41	C₂³×C₃²	—	2	2	no	—
42	C₃²×D₄	—	2	2	no	—
43	C₃²×D₄	C₂²×C₃²	1	4	yes	—
44	C₂×C₃²×C₄	C₂²×C₃²	1	1	yes	characteristic
45	C₂³×C₉	—	2	24	no	—
46	C₃²×C₈	C₂²×C₃²	1	2	yes	—
47	D₄×C₉	—	2	24	no	—
48	D₄×C₉	C₂²×C₉	1	48	yes	complemented
49	C₂×C₄×C₉	C₂²×C₉	1	12	yes	—
50	C₈×C₉	C₂²×C₉	1	24	yes	complemented
51	C₂×C₃×C₉	—	4	16	no	—
52	C₂×C₃×C₉	C₃×D₈	1	8	yes	—
53	C₂×C₃×C₉	C₂×C₃×D₄	1	4	yes	—
54	C₂×C₃×D₄	C₂×C₃×C₉	1	8	yes	—
55	C₃×D₈	C₂×C₃×C₉	1	16	yes	—
56	C₂×C₃×C₈	C₂×C₃×C₉	1	4	yes	—
57	C₂²×C₃²	—	2	4	no	—
58	C₂²×C₃²	—	4	4	no	—
59	C₂²×C₃²	C₃²×D₄	1	1	yes	characteristic
60	C₃²×C₄	C₃²×D₄	1	1	yes	characteristic
61	C₃²×C₄	C₂³×C₃²	1	1	yes	characteristic
62	C₂²×C₉	—	2	48	no	—
63	C₂²×C₉	—	4	48	no	—
64	C₂²×C₉	D₄×C₉	1	12	yes	—
65	C₄×C₉	D₄×C₉	1	12	yes	—
66	C₄×C₉	C₂³×C₉	1	12	yes	—
67	C₂×D₈	C₉²	1	1	yes	characteristic, direct product factor, Sylow
68	C₃×C₉	C₂×C₃×D₈	1	4	yes	—
69	C₂³×C₃	—	2	8	no	—
70	C₃×D₄	—	2	8	no	—
71	C₃×D₄	C₂²×C₃×C₉	1	16	yes	—
72	C₂×C₃×C₄	C₂²×C₃×C₉	1	4	yes	—
73	C₃×C₈	C₂²×C₃×C₉	1	8	yes	—
74	C₂×C₃²	—	4	4	no	—
75	C₂×C₃²	C₃²×D₈	1	2	yes	—
76	C₂×C₃²	C₂×C₃²×D₄	1	1	yes	characteristic
77	C₂×C₉	—	4	48	no	—
78	C₂×C₉	C₉×D₈	1	24	yes	direct product factor
79	C₂×C₉	C₂×D₄×C₉	1	12	yes	—
80	C₂×D₄	C₂×C₉²	1	2	yes	complemented
81	D₈	C₂×C₉²	1	4	yes	direct product factor
82	C₂×C₈	C₂×C₉²	1	1	yes	characteristic, complemented
83	C₂²×C₃	—	2	16	no	—
84	C₂²×C₃	—	4	16	no	—
85	C₂²×C₃	C₃×D₄×C₉	1	4	yes	—
86	C₃×C₄	C₃×D₄×C₉	1	4	yes	—
87	C₃×C₄	C₂³×C₃×C₉	1	4	yes	—
88	C₃²	C₂×C₃²×D₈	1	1	yes	characteristic
89	C₉	C₂×C₉×D₈	1	12	yes	direct product factor
90	C₂³	—	2	2	no	—
91	D₄	—	2	2	no	—
92	D₄	C₂²×C₉²	1	4	yes	complemented
93	C₂×C₄	C₂²×C₉²	1	1	yes	characteristic
94	C₈	C₂²×C₉²	1	2	yes	complemented
95	C₂×C₃	—	4	16	no	—
96	C₂×C₃	C₃×C₉×D₈	1	8	yes	—
97	C₂×C₃	C₂×C₃×D₄×C₉	1	4	yes	—
98	C₂²	—	2	4	no	—
99	C₂²	—	4	4	no	—
100	C₂²	D₄×C₉²	1	1	yes	characteristic
101	C₄	D₄×C₉²	1	1	yes	characteristic
102	C₄	C₂³×C₉²	1	1	yes	characteristic
103	C₃	C₂×C₃×C₉×D₈	1	4	yes	—
104	C₂	—	4	4	no	—
105	C₂	C₉²×D₈	1	2	yes	direct product factor
106	C₂	C₂×D₄×C₉²	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×C₉²

C₂

yes

maximal, complemented

C₉²×D₈

C₂

yes

maximal, direct product factor

C₂×C₈×C₉²

C₂

yes

maximal, characteristic, complemented

C₂×C₃×C₉×D₈

C₃

yes

maximal

C₂³×C₉²

—

D₄×C₉²

—

D₄×C₉²

C₂²

yes

complemented

C₂×C₄×C₉²

C₂²

yes

characteristic

C₈×C₉²

C₂²

yes

complemented

C₂×C₃×D₄×C₉

C₂×C₃

yes

—

C₃×C₉×D₈

C₂×C₃

yes

—

C₂×C₃×C₈×C₉

C₂×C₃

yes

—

C₂²×C₉²

—

C₂²×C₉²

—

C₂²×C₉²

D₄

yes

characteristic

C₄×C₉²

D₄

yes

characteristic

C₄×C₉²

C₂³

yes

characteristic

C₂×C₃²×D₈

C₃²

yes

characteristic

C₂×C₉×D₈

C₉

yes

direct product factor

C₂³×C₃×C₉

—

C₃×D₄×C₉

—

C₃×D₄×C₉

C₂²×C₃

yes

—

C₂×C₃×C₄×C₉

C₂²×C₃

yes

—

C₃×C₈×C₉

C₂²×C₃

yes

—

C₂×C₉²

—

C₂×C₉²

D₈

yes

direct product factor

C₂×C₉²

C₂×D₄

yes

characteristic

C₂×C₃²×D₄

C₂×C₃²

yes

—

C₃²×D₈

C₂×C₃²

yes

—

C₂×C₃²×C₈

C₂×C₃²

yes

characteristic

C₂×D₄×C₉

C₂×C₉

yes

complemented

C₉×D₈

C₂×C₉

yes

direct product factor

C₂×C₈×C₉

C₂×C₉

yes

complemented

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₃×D₄

yes

—

C₃×C₄×C₉

C₃×D₄

yes

—

C₃×C₄×C₉

C₂³×C₃

yes

—

C₂×C₃×D₈

C₃×C₉

yes

—

C₉²

C₂×D₈

yes

characteristic, direct product factor, Sylow

C₂³×C₃²

—

C₃²×D₄

—

C₃²×D₄

C₂²×C₃²

yes

—

C₂×C₃²×C₄

C₂²×C₃²

yes

characteristic

C₂³×C₉

—

C₃²×C₈

C₂²×C₃²

yes

—

D₄×C₉

—

D₄×C₉

C₂²×C₉

yes

complemented

C₂×C₄×C₉

C₂²×C₉

yes

—

C₈×C₉

C₂²×C₉

yes

complemented

C₂×C₃×C₉

—

C₂×C₃×C₉

C₃×D₈

yes

—

C₂×C₃×C₉

C₂×C₃×D₄

yes

—

C₂×C₃×D₄

C₂×C₃×C₉

yes

—

C₃×D₈

C₂×C₃×C₉

yes

—

C₂×C₃×C₈

C₂×C₃×C₉

yes

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₃²×D₄

yes

characteristic

C₃²×C₄

C₃²×D₄

yes

characteristic

C₃²×C₄

C₂³×C₃²

yes

characteristic

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

D₄×C₉

yes

—

C₄×C₉

D₄×C₉

yes

—

C₄×C₉

C₂³×C₉

yes

—

C₂×D₈

C₉²

yes

characteristic, direct product factor, Sylow

C₃×C₉

C₂×C₃×D₈

yes

—

C₂³×C₃

—

C₃×D₄

—

C₃×D₄

C₂²×C₃×C₉

yes

—

C₂×C₃×C₄

C₂²×C₃×C₉

yes

—

C₃×C₈

C₂²×C₃×C₉

yes

—

C₂×C₃²

—

C₂×C₃²

C₃²×D₈

yes

—

C₂×C₃²

C₂×C₃²×D₄

yes

characteristic

C₂×C₉

—

C₂×C₉

C₉×D₈

yes

direct product factor

C₂×C₉

C₂×D₄×C₉

yes

—

C₂×D₄

C₂×C₉²

yes

complemented

D₈

C₂×C₉²

yes

direct product factor

C₂×C₈

C₂×C₉²

yes

characteristic, complemented

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₃×D₄×C₉

yes

—

C₃×C₄

C₃×D₄×C₉

yes

—

C₃×C₄

C₂³×C₃×C₉

yes

—

C₃²

C₂×C₃²×D₈

yes

characteristic

C₉

C₂×C₉×D₈

yes

direct product factor

C₂³

—

D₄

—

D₄

C₂²×C₉²

yes

complemented

C₂×C₄

C₂²×C₉²

yes

characteristic

C₈

C₂²×C₉²

yes

complemented

C₂×C₃

—

C₂×C₃

C₃×C₉×D₈

yes

—

C₂×C₃

C₂×C₃×D₄×C₉

yes

—

C₂²

—

C₂²

—

100

C₂²

D₄×C₉²

yes

characteristic

101

C₄

D₄×C₉²

yes

characteristic

102

C₄

C₂³×C₉²

yes

characteristic

103

C₃

C₂×C₃×C₉×D₈

yes

—

104

C₂

—

105

C₂

C₉²×D₈

yes

direct product factor

106

C₂

C₂×D₄×C₉²

yes

characteristic

	subgroup	G/H
center	C₂²×C₉²	D₄
commutator	C₄	C₂³×C₉²
Frattini, cosocle	C₃²×C₄	C₂³×C₃²
socle	C₂²×C₃²	C₃²×D₄
prime radical	C₂×C₃²×D₈	C₃²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉² ◁ C₂×C₄×C₉² ◁ G

subgroup

G/H

center

C₂²×C₉²

D₄

commutator

C₄

C₂³×C₉²

Frattini, cosocle

C₃²×C₄

C₂³×C₃²

socle

C₂²×C₃²

C₃²×D₄

prime radical

C₂×C₃²×D₈

C₃²

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₉² ◁ C₂×C₄×C₉² ◁ G

N	H ≅ G/N	type
1. C₂×D₄×C₉²	C₂	semidirect
2. C₉²×D₈	C₂	direct (N × H)
3. C₂×C₈×C₉²	C₂	semidirect
7. D₄×C₉²	C₂²	semidirect
9. C₈×C₉²	C₂²	semidirect
19. C₂×C₉×D₈	C₉	direct (N × H)
26. C₂×C₉²	D₈	direct (N × H)
31. C₂×D₄×C₉	C₂×C₉	semidirect
32. C₉×D₈	C₂×C₉	direct (N × H)
33. C₂×C₈×C₉	C₂×C₉	semidirect
40. C₉²	C₂×D₈	direct (N × H)
48. D₄×C₉	C₂²×C₉	semidirect
50. C₈×C₉	C₂²×C₉	semidirect
67. C₂×D₈	C₉²	direct (N × H)
78. C₂×C₉	C₉×D₈	direct (N × H)
80. C₂×D₄	C₂×C₉²	semidirect
81. D₈	C₂×C₉²	direct (N × H)
82. C₂×C₈	C₂×C₉²	semidirect
89. C₉	C₂×C₉×D₈	direct (N × H)
92. D₄	C₂²×C₉²	semidirect
94. C₈	C₂²×C₉²	semidirect
105. C₂	C₉²×D₈	direct (N × H)

H ≅ G/N

type

1. C₂×D₄×C₉²

C₂

semidirect

2. C₉²×D₈

C₂

direct (N × H)

3. C₂×C₈×C₉²

C₂

semidirect

7. D₄×C₉²

C₂²

semidirect

9. C₈×C₉²

C₂²

semidirect

19. C₂×C₉×D₈

C₉

direct (N × H)

26. C₂×C₉²

D₈

direct (N × H)

31. C₂×D₄×C₉

C₂×C₉

semidirect

32. C₉×D₈

C₂×C₉

direct (N × H)

33. C₂×C₈×C₉

C₂×C₉

semidirect

40. C₉²

C₂×D₈

direct (N × H)

48. D₄×C₉

C₂²×C₉

semidirect

50. C₈×C₉

C₂²×C₉

semidirect

67. C₂×D₈

C₉²

direct (N × H)

78. C₂×C₉

C₉×D₈

direct (N × H)

80. C₂×D₄

C₂×C₉²

semidirect

81. D₈

C₂×C₉²

direct (N × H)

82. C₂×C₈

C₂×C₉²

semidirect

89. C₉

C₂×C₉×D₈

direct (N × H)

92. D₄

C₂²×C₉²

semidirect

94. C₈

C₂²×C₉²

semidirect

105. C₂

C₉²×D₈

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₄×C₉²	2
C₉²×D₈	2
C₂×C₈×C₉²	2
C₂×C₃×C₉×D₈	3

subgroup

index

C₂×D₄×C₉²

C₉²×D₈

C₂×C₈×C₉²

C₂×C₃×C₉×D₈

Subgroups — C₂×C₉²×D₈ #72244113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C92×D8 #72244113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉²×D₈ #72244113